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International Journal of Differential Equations
Volume 2014, Article ID 427547, 4 pages
http://dx.doi.org/10.1155/2014/427547
Research Article

Stability of Solutions to a Free Boundary Problem for Tumor Growth

School of Mathematics and Information Sciences, Zhaoqing University, Zhaoqing, Guangdong 526061, China

Received 5 February 2014; Accepted 14 May 2014; Published 21 May 2014

Academic Editor: Gershon Wolansky

Copyright © 2014 Shihe Xu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. M. Bodnar and U. Foryś, “Time delay in necrotic core formation,” Mathematical Biosciences and Engineering, vol. 2, no. 3, pp. 461–472, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. F. S. Borgesa, K. C. Iarosza, H. P. Renb et al., “Model for tumour growth with treatment by continuous and pulsed chemotherapy,” Biosystems, vol. 116, pp. 43–48, 2014. View at Google Scholar
  3. H. Byrne and M. Chaplain, “Growth of nonnecrotic tumors in the presence and absence of inhibitors,” Mathematical Biosciences, vol. 130, pp. 151–181, 1995. View at Google Scholar
  4. S. Fu and S. Cui, “Global existence and stability of solution of a reaction-diffusion model for cancer invasion,” Nonlinear Analysis. Real World Applications, vol. 10, no. 3, pp. 1362–1369, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. J. Ward and J. King, “Mathematical modelling of avascular-tumor growth—II: modelling grwoth saturation,” IMA Journal of Mathematics Applied in Medicine and Biology, vol. 15, pp. 1–42, 1998. View at Google Scholar
  6. J. Wu and F. Zhou, “Asymptotic behavior of solutions of a free boundary problem modeling the growth of tumors with fluid-like tissue under the action of inhibitors,” Transactions of the American Mathematical Society, vol. 365, no. 8, pp. 4181–4207, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. H. Bueno, G. Ercole, and A. Zumpano, “Asymptotic behaviour of quasi-stationary solutions of a nonlinear problem modelling the growth of tumours,” Nonlinearity, vol. 18, no. 4, pp. 1629–1642, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  8. S. Cui, “Analysis of a mathematical model for the growth of tumors under the action of external inhibitors,” Journal of Mathematical Biology, vol. 44, no. 5, pp. 395–426, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. S. Cui and A. Friedman, “Analysis of a mathematical model of the growth of necrotic tumors,” Journal of Mathematical Analysis and Applications, vol. 255, no. 2, pp. 636–677, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. S. Cui and S. Xu, “Analysis of mathematical models for the growth of tumors with time delays in cell proliferation,” Journal of Mathematical Analysis and Applications, vol. 336, no. 1, pp. 523–541, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. J. Escher and A.-V. Matioc, “Well-posedness and stability analysis for a moving boundary problem modelling the growth of nonnecrotic tumors,” Discrete and Continuous Dynamical Systems B, vol. 15, no. 3, pp. 573–596, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. J. Escher and A.-V. Matioc, “Bifurcation analysis for a free boundary problem modeling tumor growth,” Archiv der Mathematik, vol. 97, no. 1, pp. 79–90, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. J. Escher and A.-V. Matioc, “Analysis of a two-phase model describing the growth of solid tumors,” European Journal of Applied Mathematics, vol. 24, no. 1, pp. 25–48, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. U. Forys and M. Bodnar, “Time delays in proliferation process for solid avascular tumour,” Mathematical and Computer Modelling, vol. 37, pp. 1201–1209, 2003. View at Google Scholar
  15. A. Friedman and F. Reitich, “Analysis of a mathematical model for the growth of tumors,” Journal of Mathematical Biology, vol. 38, no. 3, pp. 262–284, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. S. Xu, “Analysis of tumor growth under direct effect of inhibitors with time delays in proliferation,” Nonlinear Analysis. Real World Applications, vol. 11, no. 1, pp. 401–406, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. S. Xu and Z. Feng, “Analysis of a mathematical model for tumor growth under indirect effect of inhibitors with time delay in proliferation,” Journal of Mathematical Analysis and Applications, vol. 374, no. 1, pp. 178–186, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. J. Smoller, Shock Waves and Reaction-Diffusion Equations, vol. 258, Springer-, New York, NY, USA, 2nd edition, 1994. View at Publisher · View at Google Scholar · View at MathSciNet